James Clerk Maxwell is known for his eponymous equations of electromagnetism and the velocity distribution whose naming he shares with Boltzmann. But before Maxwell studied magnetic fields and gas particles, he investigated the composition of Saturn’s Rings.

Introduction

In an essay contest in which he was both the winner and the only contestant [1][2], Maxwell modeled the composition of Saturn’s rings. He considered the implications of assuming these rings were either non-uniform, uniform but loaded with a heavy mass on one-end, composed of a discrete collection of massive satellites, entirely fluid, or a composite of multiple rings.

Maxwell’s analysis of these cases was extensive, but the simplest case was one that he did not consider since Laplace had studied it almost a century before:

The simple case:

Are the rings of Saturn uniform in mass?

This question can be answered by simple stability arguments. Saturn’s rings have existed for billions of years, and so we know it likely comprises a stable system around the planet. Thus, another way to phrase the above question, is “Is it possible for such a stable ring-planet system to have a ring of uniform density?”

Complex Saturn's Rings
Saturn's hula hoops

Heuristic Approach

First, let’s present a “back-of-the-envelope” analysis of a system with a planet and a uniform-mass ring. The system is depicted in Figure 1. We want to determine if the planet displaced radially from the center of the ring would return to the center or be pushed farther away and eventually into the ring itself, thus destroying the system.

Simple Saturn's Rings

Figure 1: Saturn's Rings depiction (Simple): Depiction of planet surrounded by uniform ring. The ring has a radius \(a\) and uniform mass density \(\lambda_0\). The planet is perturbed from the center of the ring along the (negative) \(\hat{x}\) axis.

Let the planet start in this out-of-equilibrium position. We can obtain a rough estimate of the forces acting on the planet by first considering how various sections of the ring contribute, through Newton’s Law of Gravitation, to the net force. In the figure, we labeled two (initially arbitrary) regions of the ring as I and II in order to define mass regions which contribute to the net force. By symmetry we can take the net force in the \(\hat{y}\)-direction to be zero.

Take the ring to have a uniform mass density of \(\lambda_0\), and the planet to be a radial distance \(r_\text{I}\) and \(r_\text{II}\) from the arc regions I and II, respectively. To be precise, if the planet is really displaced from the center of the circle, then \(r_\text{I}\) and \(r_\text{II}\) are not constant radial extents along their respective arc lengths. However, we can take the planet to be close enough to the center that this approximation is valid. We will also assume these arc lengths to be subtended by an angle \(\Delta \phi\), and we parameterize this angle by \(\alpha\). With all these considerations, we find that the net force on the planet in the \(\hat{x}\) direction (from the ring regions subtended by the angle \(\Delta \phi\) ) is

$$ \begin{aligned} F_{\text{Net, } x}& = F_{\text{II}\, x}(r_\text{II})- F_{\text{I}\, x}(r_\text{I})\\[.5em] & = \frac{G M_p a}{r_\text{II}^2}\int^{\Delta \phi/2}_{-\Delta \phi/2} d\alpha \,\lambda_0\, \cos\alpha - \frac{G M_p a}{r_\text{I}^2}\int^{\Delta \phi/2}_{-\Delta \phi/2} d\alpha \,\lambda\, \cos\alpha \\[.5em] & = 2G M_p\, \lambda_0 a \sin(\Delta \phi/2) \left(\frac{1}{r_\text{II}^2} - \frac{1}{r_\text{I}^2}\right) \end{aligned} $$

Taking \(\Delta \phi = \pi\) so that the entire ring is covered, we find

$$ \hspace{2cm} F_{\text{Net, } x}\Big|_{\Delta \phi = \pi} = 2G M_p\, \lambda_0 a \left(\frac{1}{r_\text{II}^2} - \frac{1}{r_\text{I}^2}\right) \hspace{2cm} (1) $$

From Figure 1, we see that if the planet is shifted in the negative \(\hat{x}\) direction, then \(r_\text{I} < r_{\text{II}}\) and by Eq. (1), \(F_{\text{Next,} x} <{0}\). We displaced the planet in the negative \(\hat{x}\) direction and found that the force was in the same direction as the displacement. Thus this force pushes the planet further away from the center (rather than back towards the center) and ultimately leads to the planet crashing into the ring itself destroying the system.

A planet perturbed from the center of a uniform mass ring (while still remaining within the plane of the ring) would experience unequal gravitational force contributions from diametrically posed arcs of the ring, and the net force would tend to push the planet farther away from the center. In other words, the system is unstable.

With this answer obtained mainly through qualitative argument and approximation, we can now turn to the more mathematically rigorous solution.

Rigorous Approach

We can derive the results of the previous section more rigorously using a Newtonian picture of the planet-ring dynamics. Although we don’t consider it here, it is also possible to obtain these results from the Lagrangian formalism.

As in the previous section, we displace the planet from the center of the ring and then calculate the force on it to determine its subsequent direction of motion. The situation is depicted in Figure 2.

Simple Saturn's Rings

Figure 2: Planet-ring system with coordinate variables. \( \rho \) is the distance from the planet to the differential length \(ds\); \(R\) is the distance from the planet to the ring's center; \(a\) is the distance from the ring's center to the differential length \(ds\); \(\phi\) is the angle between sides \(R\) and \(\rho\); \(\theta\) is the angle between sides \(R\) and \(a\). We take \(R\ll a\).

We place our coordinate system at the ring's center and assume the planet remains in the plane of the ring. By Newton's Law of Gravitation, the net force exerted on the planet by the ring is

$$ \hspace{2cm}\vec{F} = GM_{p} \int^{2\pi a}_{0} ds \, \lambda _0(s) \,\frac{\hat{\rho}}{\rho^2}, \hspace{2cm} (2) % \label{eq:F_rigor} $$

where \(s\) is the arc-length position parameter of the ring, and \(\hat{\rho}\) is a unit vector pointing from the planet towards the arc-length element \(ds\). We wrote the mass density \(\lambda_0(s)\) as a function of \(s\) for full generality, but for this system the mass density is constant.

Next, from from the law of cosines we have

$$ \rho^2 = R^2 + a^2 - 2 a R\cos\theta, $$

and from the figure, we see that the unit vector \( \hat{ \rho } \) is

$$ \hspace{2cm}\hat{\rho} = \cos\phi \, \hat{x} - \sin \phi \,\hat{y}. \hspace{2cm} (3) % \label{eq:unit_vec} $$

Forming a right triangle with an acute angle at \(\phi\), the other acute angle at \(ds\), and the right angle at a specific point to the right of the circle's center, we can show that

$$ \begin{aligned} \rho \cos\phi & = R- a\cos\theta \\ \rho \sin \phi & = a \sin\theta, \end{aligned} $$

Ss that Eq.(3) becomes

$$ \begin{aligned} \hspace{2cm} \hat{\rho} = \frac{1}{\rho} \Big[ (R- a \cos\theta) \hat{x} - a \sin \theta \,\hat{y} \Big] \hspace{2cm} (4) % \label{eq:unit_vec2} \end{aligned} $$

Plugging Eq.(4) into Eq.(2) and changing integration variables \(ds = a d\theta\), we see (since it consists of the integration of an odd function over an even domain) there is no force in the \(\hat{y}\) direction. The resulting force is only in the \(\hat{x}\) direction and has the value

$$ \begin{aligned} {\vec F} & = GM_p \int^{2\pi}_{0} a d\theta\, \lambda(\theta) \, \frac{(R-a \cos\theta)}{(R^2 + a^2 - 2 a R\cos\theta)^{3/2}}\hat{x}\\[0.5em] & = 2GM_p a\int^{\pi}_{0} d\theta\, \lambda(\theta) \, \frac{(R-a \cos\theta)}{(R^2 + a^2 - 2 a R\cos\theta)^{3/2}}\hat{x} \end{aligned} $$

where we took \(\lambda(\theta) \equiv \lambda_0 ( s= a \theta)\) and used the properties of the cosine function to cut the domain of integration by two. Making the integration dimensionless for later convenience, we have

$$ \begin{aligned} \hspace{2cm} \vec{F} = \frac{2 G M_{p} }{a} \,\frac{ I[\lambda; \delta]}{(1 + \delta^2)^{3/2}}\, \hat{x}, \hspace{2cm} (5) % \label{eq:force} \end{aligned} $$

where

$$ \begin{aligned} \hspace{2cm} I[\lambda; \delta] \equiv \int^{\pi}_{0} d\theta\, \lambda(\theta) \frac{\delta-\cos\theta}{\left(1- 2\delta \cos\theta/(1+\delta^2) \right)^{3/2}}, \hspace{2cm} (6) % \label{eq:Idef} \end{aligned} $$

and \(\delta \equiv R/a\). We write \(I[\lambda; \delta]\) to indicate that while \(I\) is a functional of the function \(\lambda\) it is a function of variable \(\delta\).

We are interested in the situation where the planet is only slightly perturbed from its center, i.e., where \(\delta = R/a \ll 1\). So, to investigate the stability of the system we should expand Eq.(5) and Eq.(6) as power series in \(\delta\). For \(I\) we have

$$ \begin{aligned} I[\lambda; \delta] & = \int^{\pi}_{0} d\theta\, \lambda(\theta) \Big( -\cos\theta + \delta (1- 3\cos^2\theta) \\ & \qquad + 3\delta^2 (\cos^2 \theta - \tfrac{5}{2} \cos^3\theta)\Big) + {O}(\delta^3). \hspace{1cm} (7) % \label{eq:Ione} \end{aligned} $$

This result is general for all \(\lambda(\theta)\) and hence could be used to investigate the orbital stability of various mass distributions. For the case of a constant mass density, we have, to lowest order in \(\delta\)

$$ \begin{aligned} I[\lambda; \delta] & = \lambda_0 \delta \int^{\pi}_{0} d\theta \, (1- 3\cos^2 \theta) + {O}(\delta^2)\\[0.5em] & = - \frac{\pi \lambda_0 \delta}{2} + {O}(\delta^2), \end{aligned} $$

where the first term in Eq.(7) integrated to zero for a constant mass density. By Eq.(5), we then have

$$ \hspace{2cm} \vec{F} = - \frac{\pi G M_p \lambda_0}{a} \left(\frac{R}{a}\right) \hat{x} + {O}(\delta^2) \hspace{2cm} (8) $$

From Figure 2, we recall that this force was computed by first assuming an initial displacement of the mass in the negative \(\hat{x}\) direction. The sign of Eq.(8) indicates that the force of the planet is in the direction of this initial perturbation, and we can therefore conclude that the planet-(uniform)-mass ring system is unstable. Therefore, Saturn's rings, which are clearly quite stable, cannot have uniform mass density.

Ending Questions

These two approaches are valuable because they illustrate how explanations in physics should succeed at multiple levels. There are often ways we can analyze systems with back-of-the envelope type calculations, but if we understand the fundamental laws and effects for such systems, our approximate and unrigorous answers should give us similar results to those from a more rigorous analysis.

How might we investigate stability for more general mass systems? In our more rigorous analysis, we computed the gravitational force, but a more direct way to investigate stability is to compute the gravitational potential (which in this case is proportional to, but not equal to, the corresponding potential energy). For this system the potential of the ring at the position of the planet is

$$ \hspace{2cm} V[\lambda; R] = -a \int^{2\pi}_{0} d\theta\, \lambda(\theta) \frac{1}{(a^2 + R^2 - 2aR\cos\theta)^{1/2}}. \hspace{2cm} (9) $$

Given \(R\ll a\) we can employ a multipole expansion of \(V\) and investigate the possible stability by the standard sequence of steps:

  1. Set \(\partial V[\lambda; R] /\partial R = 0\) and find \(R_1, \ldots, R_k\) which satisfy condition.
  2. Compute \(\partial^2 V[\lambda; R] /\partial R^2 \) at each critical point
  3. If \(\partial^2 V[\lambda; R] /\partial R^2 \big|_{R = R_{k}} > 0 \) then \(R = R_k\) is a stable critical point.

In the \(R \ll a\) limit, Eq.(9) reduces to a simple form:

$$ \begin{aligned} V[\lambda; R] & = -\int^{2\pi}_{0} d\theta\lambda (\theta) \left(1 + \delta^2 - 2\delta \cos\theta\right)^{-1/2}\\[0.5em] & = - \sum^{\infty}_{k=0} c_{k} \left(\frac{R}{a} \right)^k, \end{aligned} $$

where

$$ c_k \equiv \int^{2\pi}_{0} d\theta\, \lambda(\theta) \, P_{k}(\cos\theta) $$

and \(P_k(\cos\theta)\) are Legendre polynomials [3].

There are some natural "What if" questions that follow from all this. We chose a simple uniform density, but it is worth exploring non-uniform choices. For example, what if we took the mass density to be \(\lambda(\theta) = \lambda_0\cos^2 \theta\). Would the planet have a stable equilibrium at the center in this case? Could we devise a general statement which mass densities lead to stable equilibria?

Also, we formulated this analysis in terms of Netwoninan Mechanics, but a Lagrangian analysis is also informative. In particular, one utility of the Lagrangian formulation is that it makes it easy to incorporate the rotation of the ring into the stability analysis. Doing so gives us the equations of motion

$$ \begin{aligned} \mu \ddot{R} - \mu R \dot{\psi}^2 & = - \frac{\partial}{\partial R} U(R, \phi) \\ \mu R^2 \dot{\psi} + I (\dot{\psi}+ \dot{\phi}) & = L_{z} \\ I (\ddot{\psi} + \ddot{\phi}) & = - \frac{\partial}{\partial \phi} U(R, \phi) , \end{aligned} $$

where \(\mu^{-1} =M_{R}^{-1}+ M_{p}^{-1}\) (with \(M_R = \int^{2\pi}_{0}d\theta\, \lambda(\theta)\)), \(I = M_Ra^2\) is the moment of inertia of the ring, \(L_z\) is the conserved angular momentum, and the potential energy \(U\) is

$$ U(R, \phi) = - GM_{p}\sum^{\infty}_{k=0} \left(\frac{R}{a}\right)^{k} \int^{2\pi}_{0} d\theta\, \lambda(\theta)\, P_{k}[\cos(\theta+\phi)]. $$

The variable \(\psi\) is the angle \(\vec{R}\) (i.e., the vector from the planet to the coordinate origin) makes with the horizontal axis of the inertial coordinate system. The variable \(\phi\) defines the ring's rotation about its axis. As we expect, if we assume \(\dot{\phi} = 0\) and then \(\phi = 0\), these equations reduce to a results consistent with Eq.(5).

If we were to again assume that the ring has a constant mass density, what would the more general equations of motion above imply about the stability of the system? Does the rotational motion of the ring affect stability?

These questions and some others were explored by Maxwell in the early chapters of [2]. Since the 19th century, the understanding of Saturn's Rings has grown beyond Maxwell's theoretical investigations. We now know that the rings are composed of small satellites that orbit Saturn like cosmic vehicles in a interplanetary roundabout. Still, it is an impressive feat that long before we could send our own satellites to observe the satellites of a distant planet, it was possible to calculate through principles, logic, and lots of ink what we expected to find.

[1] Maxwell, J. Clerk. "Abstract of Professor Maxwell's paper on the Stability of Saturn's Rings." Monthly Notices of the Royal Astronomical Society, Vol. 19, p. 297-304 19 (1859): 297-304. [Web Link]

[2] S. G. Brush, C. F. Everitt, and E. Garber, "Maxwell on saturn's rings," Cambridge, MA, MIT Press, 1983, 210 p. , vol. 1, 1983.

[3] Hassani, Sadri. Mathematical physics: a modern introduction to its foundations. Springer Science & Business Media, 2013.